Convergent message passing algorithms - a unifying view

TL;DR

This paper introduces a unified framework for convergent message-passing algorithms in graphical models, demonstrating that many existing algorithms are special cases of the proposed method and providing new convergent algorithms.

Contribution

The paper presents a simple, provably convergent abstract algorithm (TCBO) that unifies many message-passing algorithms and derives new convergent variants by modifying update orders.

Findings

Many existing algorithms are instances of TCBO.

The proposed framework guarantees convergence under certain conditions.

New convergent algorithms are derived from existing non-convergent ones.

Abstract

Message-passing algorithms have emerged as powerful techniques for approximate inference in graphical models. When these algorithms converge, they can be shown to find local (or sometimes even global) optima of variational formulations to the inference problem. But many of the most popular algorithms are not guaranteed to converge. This has lead to recent interest in convergent message-passing algorithms. In this paper, we present a unified view of convergent message-passing algorithms. We present a simple derivation of an abstract algorithm, tree-consistency bound optimization (TCBO) that is provably convergent in both its sum and max product forms. We then show that many of the existing convergent algorithms are instances of our TCBO algorithm, and obtain novel convergent algorithms "for free" by exchanging maximizations and summations in existing algorithms. In particular, we show that Wainwright's non-convergent sum-product algorithm for tree based variational bounds, is actually convergent with the right update order for the case where trees are monotonic chains.